How to make eye tracker for flat eyes that already applying lattice
It is well explained here.

I don't think it is recommended to apply the Lattice modifier, and I think you should begin with a round sphere, not a flat sphere, otherwise it won't deform properly. So:


First create your lattice around your eye sphere. Keep the lattice cubic. Give your eye a Lattice modifier with the Lattice as Object. Give your lattice a basis shape key and create a second shape key.





Keep the lattice selected, go in Edit mode, select the second shape key, scale the lattice on the Y axis.





Back in Object mode, select the second shape key and push the value to 1, the eye should deform. Then bring it back to 0 for the moment.





Create a bone at the center of the eye, use the 3D cursor as Pivot Point to rotate it 90° so that it point forwards. Create a second bone that will be the target. Give the first bone a Track To constraint with this second bone as Target. Parent the eye to its bone, move the target, the eye should follow.